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Experimental and numerical investigation of sloshing resonance phenomena in a spring-mounted rectangular tank S. Pirker a,b,n, A. Aigner b, G. Wimmer c a

Christian-Doppler Laboratory on Particulate Flow Modelling, Johannes Kepler University, 4040 Linz, Austria Inst. of Fluid Dynamics and Heat Transfer, Johannes Kepler University, Linz, Austria c Siemens VAI Metals Technologies GmbH, Linz, Austria b

a r t i c l e i n f o

abstract

Article history: Received 7 January 2011 Received in revised form 15 August 2011 Accepted 12 September 2011 Available online 22 September 2011

Sloshing of liquid in a vessel can cause operational problems, especially when the characteristic frequencies of the sloshing mode coincide with the structural eigenmodes of the vessel suspension system. Inspired by sloshing phenomena in a steelmaking converter, this—predominantly experimental—study focuses on a simpliﬁed model system. In place of a complex vessel geometry, we consider a rectangular tank mounted on a spring-controlled seesaw to mimic the suspension system of the steel converter. In the course of experiments, we investigated (a) a collapsing water column in a ﬁxed tank, (b) gas injection-induced sloshing in a ﬁxed tank, (c) initial excitation-induced sloshing in a spring-mounted tank, and (d) gas injection-induced sloshing in a spring-mounted tank. In the spring-mounted tank experiments, we found that the ratio of the mechanical eigenfrequency to the suspension system and characteristic wave frequencies determines the global sloshing behaviour. When these frequencies are similar, beat-like energy transfer occurs between the wave motion within the vessel and the motion of the vessel itself. This resonance phenomenon manifests in a periodically increasing and decreasing load on the suspension system. We substantiate our experimental ﬁndings with analytical considerations and unsteady three-dimensional multiphase ﬂow simulations. The numerical predictions correlate well in principle with the experiments with respect to sloshing mode frequencies and the ﬂuid-structure resonance phenomenon, although the sloshing motion is artiﬁcially damped. & 2011 Elsevier Ltd. All rights reserved.

Keywords: Sloshing Fluid-structure interaction Free surface ﬂows Bubbly ﬂows Beating phenomenon Large eddy simulation

1. Introduction The phenomenon of sloshing in a vessel partially ﬁlled with liquid has great importance in cases such as tanks in ships and large road and airspace vehicles, and in evaluating the seismic response of water containers (Rebouillat and Liksonov, 2010). In road tankers, sloshing can interfere with driving dynamics and cause safety concerns (Aliabadi et al., 2003). Therefore, considerable research effort focuses on reducing the effect of sloshing on the vessel and its supporting system by introducing bafﬂes (e.g., Warnitchai and Pinkaew, 1998) or screens (e.g. Faltinsen and Timokha, 2010a). This paper was inspired by a sloshing resonance phenomenon in a steelmaking converter (Fig. 1a) in which it is believed that high oscillation loads arising from sloshing bath movement cause increased wear of the support bearings. According to previous

n Corresponding author at: Johannes Kepler University, Christian-Doppler Laboratory on Particulate Flow Modelling, 4040 Linz, Austria. E-mail address: [email protected] (S. Pirker).

0009-2509/$ - see front matter & 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.ces.2011.09.021

studies (Fabritius et al., 2005), the wave motion may be the result of intensive submerged gas injection into the vessel. Rising gas bubbles are believed to trigger an unsteady wave pattern, which, in turn, excites the mechanical converter suspension system at its characteristic frequency (Fig. 1c). From a physical point of view, the phenomenon of sloshing can be closely linked to the behaviour of surface gravity waves and their interaction with moving or ﬁxed container walls. Sloshing can be either smooth, with a well-deﬁned interface between the liquid and the covering gas phase, or violent, with disintegration of the free surface, breaking waves, and subsequent droplet formation. While in smooth sloshing the wave pattern can be described analytically on the basis of linear or non-linear wave theory (e.g., Landau and Lifshitz, 1987; Frandsen, 2004; Faltinsen et al., 2005; Faltinsen and Timokha, 2010b), these approaches fail in cases of violent sloshing. Simulations based on ﬁnite volume methods have been successfully applied to predict sloshing phenomena (e.g., Liu and Lin, 2008; Nandi and Date, 2009; Chen et al., 2009; Eswaran et al., 2009, and literature cited therein). Godderidge et al. (2009) showed that in the case of violent sloshing the gas phase should

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Fig. 2. Sketch of the simpliﬁed spring-mounted tank system; (1) evacuation system, (2) levitation rod for the vacuum box, (3) vacuum box with water column, (4) tank, (5) spring, (6) load cells, (7) gas ﬂow metering, (8) gas injection nozzle, and (9) laser distance measurement.

Normalized moment [-]

1 0.5 0 -0.5 -1

0

5

10

15

20

Time [s] Fig. 1. Sketch of (a) a steelmaking converter suspension system with (b) a torsion rod and (c) a typical monitor of the moment of this torsion rod.

also be reﬂected as a component of a multiphase ﬂow simulation. In this context, Smoothed Particle Hydrodynamics (SPH) methods have also proven to be suitable modelling candidates (Cleary et al., 2007; Delorme et al., 2009). In most experimental studies in the literature, a sloshing motion is induced by a speciﬁc tank movement (e.g., Khosropour et al., 1995; Liu and Lin, 2008; Chen et al., 2009; Godderidge et al., 2009). Attari and Rofooei (2008) placed a ﬂexible beam system between an oscillating ground plate and the tank in order to mimic the effect of earthquakes on an elevated water storage tank. Nasar et al. (2008) studied sloshing in a tank mounted on a barge subjected to global waves. Whereas in most cases a rigid and non-deformable wall material is assumed, Eswaran et al. (2009) considered elastic—and ¨ et al. (2008) even plastic—wall deformations. Akkose Our work differs in two aspects from the approaches mentioned above. Firstly, the sloshing waves are not triggered by tank

movement but by gas injection into the bath. Secondly, the tank motion is not caused by an outside source (e.g., an oscillating ground plate), but results from the interaction between the sloshing load on the tank and the dynamics of the tank suspension system. In the case of a steel converter, the vessel can rotate around its suspension axis. Depending on its tilting angle, the converter’s suspension system tries to reinforce the original tank orientation by a rotational spring-like torsion rod (Fig. 1a and b). This paper investigates sloshing phenomena and resulting mechanical loads observed in steel converters by means of a simpliﬁed substitute system. The complex vessel shape is reduced to a rectangular tank, and the complex converter suspension system is modelled by a spring-controlled seesaw (Fig. 2). In our experiments, we evaluated general sloshing characteristics by visual observation, sloshing load by load cells monitors, and tilting angle by laser distance measurements. The main focus of this study is on experimental investigation, while time-dependant three-dimensional multiphase Volume of Fluid (VOF) simulations were conducted only for selected cases in order to demonstrate their applicability and predictability. The next section presents the simpliﬁed experimental setup, followed by the numerical modelling in Section 3. Section 4 provides a comparative evaluation of the analytical, experimental, and numerical results, detailing the collapsing water column experiments, gas injection-induced sloshing, and initial excitation-induced sloshing. Finally, a conclusion and outlook are provided.

2. Experimental setup The complex original vessel shape was reduced to a rectangular tank. The length to depth ratio of the tank was chosen as a/b¼3/1 such that, according to linear wave theory (described in the next section), the frequency of the third long side sloshing mode coincides with that of the ﬁrst lateral mode. The acrylic glass tank was ﬁlled with distilled water, and throughout all the experiments both water and ambient air temperature were approximately 20 1C.

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Depending on the experimental procedure, the tank was placed either directly on a rigid frame or on a spring-controlled seesaw (Fig. 2). The ﬁrst set of experiments was conducted in a non-moving tank (Sections 4.1 and 4.2), while in the second set the tank was allowed to rock on the seesaw (see Sections 4.3 and 4.4), which itself was mounted on a rigid frame. In all cases, the rigid frame was levelled by four load cells (of type PW6DC3 from HBM). Obviously, this suspension system is over-determined in that the distribution of the weight of the tank across the two load cells on each side is not well deﬁned. In the subsequent signal evaluation, the left and right signals were therefore summed in order to detect pure long side sloshing motion. In the moving tank experiments, the tilting angle was additionally recorded by laser distance measurement. The collapsing water column in the ﬁrst experiment (see Section 4.1) was produced by a vacuum box (Fig. 2). The square box was positioned off-centre at one side of the tank just above the surface so that there was no air connection between the interior of the box and the environment. Next, the air inside the box was sucked out by a vacuum device (of type ZU05S from SMC), which raised the water level inside the box. When the box is lifted manually, a water column collapses into the main bath and induces a reliably repeatable sloshing motion. In the case of gas injection-induced sloshing (see Sections 4.2 and 4.4), pressurised air was introduced into the bath via a nozzle with a diameter of dn ¼2 mm. The nozzle tip was positioned at a distance of 115 mm from the wall and 20 mm from the tank bottom (Fig. 2). The gas ﬂow was directed horizontally at the tank centre and could be adjusted via a valve that was, in turn, controlled by a metering oriﬁce. In this study, a constant gas ﬂow rate of V_ g ¼ 1:25 103 m3 =s was used. The spring-controlled seesaw suspension system used to control the tank motion in the case of ﬂuid–solid sloshing interaction (see Sections 4.3 and 4.4) included two springs that can be placed at a distance on each side of the pivot of the seesaw. Depending on the spring constant, the spring distance, and the tank ﬁll level (Fig. 2), this suspension system can be described adequately by ð1Þ

with j denoting the tilting angle of the tank. The tilting moment Mx depends on the spring position and characteristics, and obeys M x ¼ F sp Lsp ¼ 2ksp L2sp tanðjÞ,

that at walls the normal velocity component must be zero, i.e., vn9wall ¼(qj/qn)¼ 0, the velocity potential j describing a standing wave pattern obeys

j ¼ A cosðo tÞcoshðkðz þhÞÞ:

ð3Þ

Here, o is the circular frequency, k is the wave number, h is the mean ﬁll level, and A denotes the amplitude of the wave. From the kinematic boundary condition at the free surface, we ﬁnd the relation between k and o to be

o2 ¼ gktanhðkhÞ,

ð4Þ

with g as the gravitational acceleration acting in z direction. For standing waves inside a rectangular tank, the wave numbers satisfy the further constraint 2 m n2 2 k ¼ p2 þ 2 , ð5Þ 2 a b with a and b being the lateral tank dimensions. Variables m and n describe the wave mode numbers in both lateral directions. 3.2. Numerical modelling Since the analytical approach above fails in cases of violent sloshing and additional gas injection, an additional three-dimensional numerical simulation was run. The Volume of Fluid (VOF) model (Hirt and Nichols, 1981) seemed to be a suitable modelling candidate to resolve the free surface motion of the sloshing waves. In principle, this model is based on a method in which a marker—the volume fraction—decides whether a computational cell is ﬁlled with liquid or with gas. The distribution of these markers determines the position of the stratiﬁed ﬂuids and can be deduced from a set of phase conservation equations: @aq rq @t

þ rðaq rq uÞ ¼ 0,

with the phases volume fractions aq summing up to one: X aq ¼ al þ ag ¼ 1:

ð6Þ

ð7Þ

q

2

d j M x ¼ Ixx 2 , dt

145

ð2Þ

with ksp denoting the linear spring constant and Lsp denoting the distance between spring and pivot of the seesaw. For small disturbances, Eq. (2) can be further simpliﬁed by assuming that tanðjÞ j. The torque of inertia Ixx accounts for the inertia of both the tank and the ﬂuid mass inside, and thus increases with higher ﬁll levels. Before embarking on the sloshing experiments, we checked the above model of the suspension system by manually exciting the empty tank and recording the subsequent load cell signals. The set of measurement signals (i.e., four load cell signals, the laser distance signal, and the gas ﬂow metre signal) were collected by a data acquisition system (of type Spider 8 from HBM) with a sampling rate of 1000 Hz. Furthermore, all experiments were video-monitored (selected cases even with high-speed video).

3. Analytical considerations and numerical modelling 3.1. Analytical considerations For small excitations, the sloshing wave pattern in a rectangular tank can be described by linear wave theory based on the potential ﬂow assumption (Landau and Lifshitz, 1987). Assuming

In above equations, the variables rq denote the densities of the ﬂuids. Indices l and g refer to the liquid and the gas phase, respectively, while index q refers to a general phase. The velocity u of the mixture of ﬂuids is based on a momentum equation for the mixture: @ðruÞ þ rðruuÞ ¼ rp þ rs þ rg, @t with the phase-averaged density X r¼ aq rq :

ð8Þ

ð9Þ

q

In Eq. (8), p denotes the pressure, g is the vector of gravitation, and s is the stress tensor that, in the case of violent sloshing, includes the effects of unresolved turbulent vortices. To account for the motion of the tank, an Arbitrary Lagrangian Eulerian (ALE) approach was applied. The computational grid, which is ﬁxed to the non-deformable tank geometry, is moved in accordance with the dynamic suspension system of the tank (Eq. (1)). As a consequence, the mixture velocity u in Eqs. (6) and (8) must be replaced by uugr, with ugr being the local grid velocity. At this point, the above set of conservation equations includes two mass balances for both phases and one common momentum balance for the mixture. If these equations are applied without special treatment of the interfaces, the boundaries between the phases become indistinct and the phases run into each other. Thus, an interface sharpening algorithm in a second step is required. For this purpose, we chose a stepwise linear reconstruction of the

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surface (Young, 1982) because it keeps stratiﬁed ﬂuids well separated. Sloshing motion can be laminar or turbulent. For laminar ﬂow, the stress tensor in Eq. (8) simply obeys

s ¼ mðru þ ruT Þ,

ð10Þ

with m denoting the molecular dynamic mixture viscosity X m¼ aq mq :

ð11Þ

q

For turbulent sloshing, the stress tensor must be modiﬁed. In the course of this study, we tested both Unsteady Reynolds Averaged Navier Stokes (URANS) turbulence models and Large Eddy Simulations (LES). Since, in our context, the URANS approaches did not correctly model violent gas injection-induced sloshing, we present only the LES approach. Here, the molecular mixture viscosity m is augmented by a sub-grid viscosity msg, which accounts for the effect of unresolved turbulent vortices. The standard Smagorinsky assumption (1963)

msg ¼ 2ðC s Dgr Þ2 g_ ,

ð12Þ

with Dgr denoting the local grid spacing, Cs ¼ 0.1 being a model constant, and g_ denoting the second invariant of the shear rate tensor, was applied as a ﬁrst-guess sub-grid model. In gas injection-induced sloshing, the interaction between the rising bubble plume and the liquid ﬂow must be modelled very carefully. The prospective model should be able to represent highly turbulent bubble ﬂows with signiﬁcantly varying bubble volume fractions. Furthermore, the model of the interaction between the rising bubbles and the surrounding liquid should be applicable to various ﬂow regimes and phase pairings. We chose a combination of the drag models of (i) Ishii and Zuber (1979), to describe bubbles in liquid, and (ii) Schiller and Naumann (1935), to model droplets in gas, as our primary modelling candidate (Singh, 2010). This model synthesis is based on phase-speciﬁc length scales, Lg ¼

4 r 4 r a d ,L ¼ a d , L ¼ bLg þ ð1bÞLl , 3 l b rl l 3 b d rg

ð13Þ

with db and dd denoting the mean bubble and droplet diameters, respectively, and b denoting a coupling parameter: 0:7ab : ð14Þ b ¼ min 1,max 0, 0:5 If the bubble phase were modelled in an Eulerian frame of reference, ab would be equal to ag. In our case of Lagrangian droplet tracing, ab must be determined by monitoring the number of droplet trajectories passing an individual Eulerian grid cell. For brevity, we omit details of the standard drag models of Ishii and Zuber (1979) and Schiller and Naumann (1935). Both models provide expressions for the drag coefﬁcient CD, which must be merged according to C D ¼ bC D,Ishii þ ð1bÞC D,Schiller :

ð15Þ

The volumetric interaction force between the dispersed bubble ˇ phase and the continuous liquid phase obeys (e.g., Strubelj et al., 2009) F drag ¼ C D r

al ab L

9uub 9ðuub Þ

ð16Þ

in an Eulerian frame of reference. In a Lagrangian frame of reference, this Eulerian phase interaction force must be translated into speciﬁc drag forces acting on individual bubbles in each computational cell according to X mb,i F drag V c ¼ f , ð17Þ Nb,i drag,i i

with Vc denoting the cell volume, Nb,i the number of physical bubbles in a computational parcel, and mb,i the individual mass of a bubble. In our case, the global interaction force is equally distributed across the individual bubbles or droplets that are currently present in the cell. This relation guarantees that the same momentum transfer is realized in a Lagrangian droplet tracing approach as in a corresponding Eulerian bubble phase approach. The bubbles form at the submerged nozzle with a given initial velocity and diameter, and are deleted when they reach the surface of the bath. Numerical simulations were performed partly with OpenFoam (Jasak, 1996) and partly with Fluent (2006). In the latter case, Fluent’s Discrete Phase Model (DPM) was adapted by userdeﬁned functions in order to account for high bubble volume fractions. Typical mesh sizes range between 100 k and 500 k hexahedral cells. In addition, a dynamically adapted grid reﬁnement was applied in the area of the free surface. For non-moving tank geometry, grid independence was achieved by stepwise reﬁning of the basic grid until the resulting characteristic wave frequencies remained constant. In order to avoid creeping mass losses in the VOF simulation, global masses of each phase (water and air) were monitored during each simulation. When creeping mass defects were detected, global mass balance was maintained by triﬂing artiﬁcial mass sources.

4. Results In the following section, we describe our set of experiments and carefully evaluate the results. Furthermore, we compare these experimental ﬁndings with analytical predictions and numerical simulations. Nevertheless, the focus of our research is on the experiments, while analytical considerations provide further physical interpretations, and numerical simulations conﬁrm their applicability to more complex geometries and alternative phase pairings in future experiments. In a steelmaking converter, a typical time-dependant load signal, which can be obtained from strain gauges, sometimes exhibits a characteristic amplitude modulation. If this type of resonance occurs, the magnitude of the corresponding load signal increases and decreases periodically (Fig. 1c). In this signal, the primary characteristic frequency of 2 Hz is signiﬁcantly higher, and the secondary characteristic frequency of 0.2 Hz is signiﬁcantly lower, than would be expected by evaluating the ﬁrst sloshing mode either analytically by assuming a substitute cylindrical geometry or by running a simple numerical simulation of vessel tilting. Furthermore, this resonance phenomenon occurs only in speciﬁc plant geometries and under speciﬁc process conditions. Consequently, we hypothesise that, in this case, a higher sloshing mode might be excited at a frequency that is very close to the structural frequency. The following set of experiments aimed to characterize this phenomenon. 4.1. Dam break-induced sloshing in ﬁxed tank In the ﬁrst set of experiments, the springs shown in Fig. 2 were replaced with rigid rods to obtain a non-moving, ﬁxed tank. On one side of the tank, a vacuum pump sucked a water column into a square box. When this box is lifted, the water column collapses into the tank and initiates a travelling wave. After a couple of wave reﬂections, a standing wave motion can eventually be observed. Fig. 3a shows a typical time-dependant signal of the load cells, which indicates a slowly decaying sloshing motion. In the frequency space, data from the load cell signals can easily be

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Fig. 3. Experimentally (left) and numerically (right) obtained monitors for torque (a–d) and tilting angle (e–h) resulting from (a,b) a collapsing water column in ﬁxed tank, (c,d) gas-induced sloshing in ﬁxed tank, (e,f) initial excitation-induced sloshing in spring-mounted tank, and (g,h) gas injection-induced sloshing in spring-mounted tank; all values were normalised by their maximum.

compared with the analytical predictions described in Section 3.1. In Fig. 4a, the analytically predicted frequencies for the sloshing modes in longitudinal directions are indicated by vertically slashed lines. Obviously, only the uneven modes are excited in an asymmetric water column collapse. In general, it can be observed that the experimentally obtained characteristic frequencies agree very well with the analytical predictions. Also, the numerical simulation results of the collapsing water column experiment correlate well with the experimentally obtained results (Figs. 3b and 4b).

4.2. Gas injection-induced sloshing in ﬁxed tank The second experimental conﬁguration also considers a non-movable tank. Off-centre gas injection results in load cell monitor signals as shown in Fig. 3c. In Fig. 4c, these signals are evaluated in the frequency space. Although in this case the experimentally obtained signals are degraded by higher order disturbances, the experiments clearly show that the analytically predicted characteristic sloshing modes can also be excited by gas injection.

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Fig. 4. Experimentally (left) and numerically (right) obtained frequencies for torque (a–d) and tilting angle (e–h) resulting from (a,b) a collapsing water column in ﬁxed tank, (c,d) gas-induced sloshing in ﬁxed tank, (e,f) initial excitation-induced sloshing in spring-mounted tank, and (g,h) gas injection-induced sloshing in spring-mounted tank; all values were normalised by their maximum; in addition, the analytically obtained characteristic sloshing frequencies (red) and the structural eigenfrequency (green) are indicated by vertical lines. (For interpretation of the references to colour in this ﬁgure legend, the reader is referred to the web version of this article.)

Fig. 3d presents the corresponding numerically predicted load cell signals. Obviously, the multiphase simulation of the bubble plume smears out higher-order disturbances. This is most probably caused by artiﬁcial damping due to numerical diffusion in combination with an insufﬁciently resolved grid of 400 k cells. Since a signiﬁcant increase in grid cells would not have been possible with the given resources, we accepted these results despite their short-comings. Prospective alternative remedies are given in the outlook of this paper. An evaluation of the numerical results in the frequency space (Fig. 4d) exhibits that the primary characteristic sloshing frequencies can also be identiﬁed with this complex multiphase simulation. 4.3. Initial tilting-induced sloshing in a spring-mounted tank In this experimental conﬁguration, the two springs were mounted such that a characteristic mechanical eigenfrequency

close to that of the the third sloshing mode was excited. This was achieved by simply exciting the empty tank manually. At the beginning of a sloshing experiment, the tank was tilted manually by about 5 degrees. Before release, the tank was kept in this position until the water surface was smooth. After release, the sloshing motion produced a load cell signal as shown in Fig. 3e. Obviously, this setting produced a frequency modulation with a primary characteristic frequency that is higher, and a secondary characteristic frequency that is lower, than the corresponding ﬁrst sloshing mode frequency. Watching this experiment one can clearly see that at ﬁrst the third sloshing mode was excited until after approximately 6 s the water surface levels out before the wave motion is fuelled again (see Movie ‘Sloshing.MP4’ in the online version of this paper). During the period the sloshing motion is nearly completely levelled out, the tank motion was intensiﬁed. Obviously, energy is transferred periodically between

S. Pirker et al. / Chemical Engineering Science 68 (2012) 143–150

the sloshing waves and the mechanical system in this resonance phenomenon. Supplementary material related to this article can be found online at doi:10.1016/j.ces.2011.09.021. In the frequency space it can be observed that, for signiﬁcant amplitude modulation, the mechanical eigenfrequency is similar to the third sloshing mode frequency. If the mechanical and the sloshing frequencies are not similar, this frequency modulation phenomenon does not occur. Fig. 5 shows the results of this initial tilting experiment in relation to the ﬁll level of the tank. It can be shown that, with the same mechanical settings, the frequency resonance phenomenon occurs only under speciﬁc operating conditions. For a ﬁll level of 100 mm, resonance is most pronounced, while for a ﬁll level of 50 mm, no resonance occurs at all. This can be readily explained by the analytical prediction that the sloshing frequencies rise with increasing ﬁll levels of the tank. At the same time, the inertia of the mechanical system is increased by the additional water mass, which results in reduction of the mechanical eigenfrequency. In our experiment, both frequencies (the mechanical eigenfrequency and

149

the third sloshing mode frequency) converge with stepwise increase of the ﬁll level until the resonance phenomenon is most pronounced. At further increase, the sloshing frequency exceeds the mechanical eigenfrequency, and the trend is reversed (Fig. 5). Fig. 4f shows that the numerically predicted resonance frequencies agree well with those in the experiments. Again, the simulations suffer from artiﬁcial damping. While the experimentally observed sloshing motion persists for more than 2 min, the numerically predicted sloshing is completely levelled out after about 20 s (Fig. 3e and f). In further experiments, we found that also the ﬁfth and the seventh sloshing mode can lead to frequency modulation if the mechanical system is adapted accordingly. For brevity, we omit these results in this paper. 4.4. Gas injection-induced sloshing in a spring-mounted tank In this ﬁnal experimental conﬁguration, gas was injected into the spring-mounted tank. The springs were arranged in the same

Fig. 5. Experimentally obtained monitors of tilting angle (left) and corresponding frequencies (right) for initial excitation-induced sloshing in spring-mounted tank against the ﬁll level (top to bottom: ﬁll level is 50, 100, 150, and 200 mm); all values were normalised by their maximum; in addition, the analytically obtained characteristic sloshing frequencies (red) and the structural eigenfrequency (green) are indicated by vertical lines. (For interpretation of the references to colour in this ﬁgure legend, the reader is referred to the web version of this article.)

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way as in the previous experiment. That is, the mechanical eigenfrequency of the tank with a ﬁll level of 150 mm was chosen close to the third sloshing mode frequency in order to promote the above-described resonance phenomenon. However, in this case sloshing was not induced by initial tilting but by eccentric gas injection, as described in Section 4.2. Fig. 3g shows a typical signal of the load cell. In gas-induced sloshing, the above-mentioned resonance phenomenon can also be observed, although it is not as regular as in initial tiltinginduced sloshing. Fig. 4g illustrates that, in this case, the peaks in the frequency space are markedly less distinct. In gas-induced sloshing, the occurrence of resonance sloshing also depends on the ﬁll level of the tank. Corresponding numerical simulations also show a resonance phenomenon, as depicted in Figs. 3h and 4h. Similar to those in the experimental ﬁndings, the resulting peaks in the frequency space are less distinct. However, the numerically predicted level of bath excitation is lower than in the experiment. This is in line with the outcome of previous simulations and can, again, be explained by dominant artiﬁcial damping.

5. Conclusion and outlook In this predominantly experimental study, the sloshing behaviour of melt inside a steel converter was investigated by means of a simpliﬁed, rectangular tank conﬁguration. In the course of a set of experiments, the sloshing motion was studied both in a non-moving and in a spring-suspended tank system. Sloshing waves were excited by a collapsing water column, manual tank movement, and by gas injection. Additional analytical considerations and numerical simulations further round off the experiments. The main ﬁndings of this study can be summarised as follows:

The resonance sloshing behaviour observed empirically in

steel converters can be reproduced qualitatively by the proposed simpliﬁed tank conﬁguration. The main reason for the occurrence of this resonance frequency modulation is that the eigenfrequency of the tank suspension system is similar to a characteristic higher sloshing mode frequency. It was also demonstrated that the occurrence of resonance depends on the ﬁll level. In the rectangular tank conﬁguration, the characteristic sloshing frequencies can be predicted well by linear wave theory, although this approach delivers no information regarding the amplitude of the individual sloshing modes. All experimental conﬁgurations can be modelled in principle by means of unsteady, three-dimensional multiphase simulations. The agreement between the numerically predicted characteristic sloshing frequencies and those in the experiment is generally good. The numerical simulations are able to model the frequency modulation phenomenon both for manual excitation and for gas-induced sloshing. The ﬁnite volume approach we chose, however, inherently suffers from numerical diffusion, resulting in artiﬁcial damping of the sloshing motion.

In order to address the problem of numerical damping, we will test alternative simulation approaches, such as Smoothed Particle Hydrodynamics (SPH), which we will include in our group’s open

source Discrete Element Model (DEM) software called LIGGGHTS (www.liggghts.com, Kloss and Goniva, 2010).

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